Preventing bias by not providing irrelevant data
Why can't we just feed the training data without data that we would consider discriminatory or irrelevant, for example, without fields for gender, race, etc., can AI still draw those prejudiced connections? If so, how? If not, why has this not been considered before?


Yes. The AI/ model still can learn those prejudiced connections. Consider that you have a  third variable which is a confounding variable or has spurious relationship that is correlated with the bias variable (BV) and the dependent variable (DV). And, the analyst removed the BV but failed to remove the third variable from the data that is fed to the model. Then the model will learn the relationships the analyst didn't want it to learn.

But, at the same time the removal of the variables could lead to omitted variable bias, which occurs when a relevant variable is left out.

Ex:

Suppose that the goal is prediction of salary ($S$) of an individual and the independent variables are age ($A$) and experience ($E$) of the individual. The analyst wants to remove the bias that could come in because of age. So, she removes age from one of the models and comes up with two competing linear models:

$S = \beta_0 + \beta_1E + \varepsilon$

$S = \beta_0 + \beta_1^*E + \beta_2A + \varepsilon$

Since, experience is highly correlated with age, in presence of age in the model, it is very likely that $\beta_1^* &lt; \beta_1$. $\beta_1$ will be a bogus estimate of a person's experience on salary as the first model suffers from the omitted variable bias. 

At the same time the predictions from the first model would be reasonably good although the second model is very likely to beat the first model. So, if the analyst wants to remove any 'bias' that might come in because of age i.e. $A$ she must also remove $E$ from the model.